524 Dr. J. It. Airey on Bessel and Neumann 



Taking the first integral, 



if 00 , . w ifYi Khf;2 * 4 ™ 4 \ I- W 



— I coskw cos^(sin w— w)dw—. - \ (1 ^- + -~- ... jcos z (smiv — w) dw 



As in the preceding example, 

 y" o cost (An w-w)dw= 2^73 [(f)* r(|) ~Wo& F (*)~ -]■ 



Similarly, 



_ ij , *? cos * (sin w ~ "° dw = 2^v3 & (!)' r (b) + 4s 6/ r (a) ' •] 



and so on. 

 The terms arising out of! the second integral of (12) can 

 be found in a similar way from 



1 x v - ] sin x dx = T(p) sin^- . 

 * o 

 Collecting the terms, the following expression for J n (z) is 

 obtained, when z=n + tc : — 



-K 6 )MiM«.©* r ©+¥ B .©MI) 



-P-(S?r&] '■ • • 4 



where 

 B =l, 



Bj= K, 



^l" 15' 



K* K 



v. — _ Jr. _l 

 ^^ 24 ~~ 24 + 280' 



B. 



1 



T ,)OU U^i 



720 1440"*" 288 3600' 



B,= 



()A/i "t" 



19* 



13* 



5010 - 900 "*" 12600 31500 



Bo = 



+ 



71* 3 



121 



+ « 



7939* 



362880 ~ 30240 + 604800 907200 + 23284*000 



